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Full text of "Mathematical And Physical Papers - Iii"

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which is analogous to the equation of continuity of an. incompressible fluid.
If we suppose all possible systems of values assigned in succession to the constants JD1? EI} Flt the formulae (6) will express all possible modes of transfer, consistent with our original assumption respecting the forms of fxyfy} fz, by which the state of temperature of the solid at the end of the time t can pass into its state at the end of the time 14- dt. Of course, if we suppose heat to be material, we cannot help attaching to it the idea of individuality. But if we suppose heat to consist in motion of some sort, which for my own part I regard as by far the more probable hypothesis, we require a definition of sameness of heat, supposing we find it convenient to treat the subject in this way. I am not now going to follow any further the subject which has just been broached; but I thought it might be worth while to point out in what manner the additional arbitrary constants found in the general expressions for the flax beyond what appeared in the equation of motion, or more properly the equation of successive distribution, corresponded to an attribute of heat which is necessarily involved in the idea of a flux, but which is not necessarily involved in the idea of the successive distributions of a given quantity of heat.
9. Besides the general equation (5), it is requisite to form the equation of condition which has to be satisfied at the surface when the solid radiates into a space at a different temperature. Let P' be a point in the surface, dS an element of the surface surrounding P', P'N a normal drawn outwards at P', P a point in NPr produced, situated at the distance 8 from P'. Consider the element of the solid bounded by dS, by a plane through P parallel to the tangent plane at P, and by a cylindrical surface circumscribing dS, and having its generating lines parallel to P'N'j and suppose this element to be indefinitely diminished in such a manner that S vanishes compared with the linear dimensions of dtt, which is allowable if the curvature at P' be finite (that is, not infinitely great), as it must necessarily be in general. The quantity of heat which enters the element across the plane through P, as well as the quantity which escapes across dS, varies ultimately as dS. The area of the cylindrical surface varies as 8 multiplied by the perimeter of dS, and there-